Class 8 NCERT Solutions – Chapter 2 Linear Equations in One Variable – Exercise 2.6

We use cross multiplication in this exercise a lot of times, so it is explained here before.

Let,
a/b = c/d
Now if we multiply both sides by the denominators of left side and right side, we get,
(a/b) X (b X d) = (c/d) X (b X d)
=> a X d = b X c
This is called cross multiplication.

Question.1 Solve the following equations.

We can solve the problems 1 to 5 by trying to bring all the unknown variables to the left side.

1.  (8x-3) / 3x = 2

Solution:

By multiplying on both sides by 3x we get,
=> (8x-3) X (3x) / 3x = 2 X (3x)
=> 8x-3 = 6x
=> 8x-6x-3 = 0
=> 2x-3 = 0
=> 2x = 3
=> x = 3/2

2. 9x / (7-6x) = 15

Solution:

By multiplying both sides by (7-6x) we get,
=> (9x) X (7-6x) / (7-6x) = 15 X (7-6x)
=> 9x = (15 X 7) – (15 X 6)x
=> 9x = 105 – 90x
=> 9x + 90x = 105
=> 99x = 105
=> x = 105/99
=> x = 35/33

3.  z / (z+15) = 4 / 9

Solution:

By cross multiplication,
=> z X 9 = (z+15) X 4
=> 9z = 4z + 4 X 15
=> 9z – 4z = 60
=> 5z = 60
=> z = 60/5
=> z = 12

4.  (3y+4) / (2-6y) = 2 / 5

Solution:

By cross multiplication,
=> (3y+4) X 5 = (-2) X (2-6y)
=> (5 X 3)y + (4 X 5) = (-2 X 2) + (-2 X -6)y
=> 15y + 20 = -4 + 12y
=> 15y -12y = (-4) + (-20)
=> 3y = -24
=> y = -24/3
=> y = -8

5.  (7y+4) / (y+2) = – 4 / 3

Solution:

By cross multiplication,
=> (7y+4) X 3 = -4 X (y+2)
=> (7 X 3)y + (4 X 3) = -4y + (-4 X 2)
=> 21y + 12 = (-4y) + (-8)
=> 21y + 4y = (-8) + (-12)
=> 25y = -20
=> y = -20/25
=> y=-4/5

6.  The ages of Hari and Harry are in the ratio 5:7. Four years from now the ratio of their ages will be 3:4. Find their present ages.

Solution:

Let the present age of Hari be x ,
Let the present age of Harry be y .

Presently their ages are in ratio 5:7 , So we get
=> x : y = 5 : 7
We get,
=> x/y = 5/7
By cross multiplication,
=> 7x = 5y
=> x = (5y) / 7 ……….. (1)

After 4 years,
Hari’s age will be x+4,
Harry’s age will be y+4.

The ratio between their ages after four years is 3:4. So we get,
=> (x+4) : (y+4) = 3 : 4
=> (x+4)/(y+4) = 3/4
By cross multiplication,
=> (x+4) X 4 = 3 X (y+4)
=> 4x +16 = 3y + 12
=> 4x – 3y = -4 ……….. (2)

Now , we got two euations.
x = (5y) / 7 ……….. (1)
4x – 3y = -4 ……….. (2)

If we substitute this x value from (1) in equation (2) we get
=> 4 X (5y/7) – 3y = -4
=> 20y/7 – 3y = -4
=> 20y/7 – (7X3)y/7 = -4
=> (20y -21y) / 7 = -4
=> -y/7 = -4
=> y = (-4) X (-7)
=> y = 28

By substituting y=28 value in (1) we get
=> x = (5 X 28) / 7
=> x = (5 X 4)
=> x =20

So here Hari’s present age is 20 years and
Harry’s present age is 28 years.

7.  The denominator of a rational number is greater than its numerator by 8. If the numerator is increased by 17 and the denominator is decreased by 1, the number obtained is 3/2. Find the rational number.

Solution:

Let the numerator be x,
and the denominator be y.

From the first part of the question we get,
=> denominator = numerator + 8
=> y = x + 8 ……….. (1)

Now, from the second part of the question ,
=> (x+17) / (y-1) = 3/2
By cross multiplication,
=> (x+17) X 2 = 3 X (y-1)
=> 2x + 34 = 3y – 3
=> 2x – 3y = -34 – 3
=> 2x – 3y = -37 ……….. (2)

We got two equations.
Substituting (1) in (2), we get
=> 2x − 3 X (x+8) = −37
=> 2x − 3x − 24 = −37
=> 37 − 24 = x
=> x = 13

By substituting x=13 in (1) we get,
=> y = 13 +8
=> y = 21

We got x=13 and y=21
Hence the original rational fraction will be 13/21